The finite element method is known from the “Dubbel - Taschenbuch für den Maschinenbau” [Dubbel's Handbook of Mechanical Engineering], 20th Ed., Springer Verlag, 2001, C48 through C50, from B. Klein: “FEM - Grundlagen und Anwendungen der Finite-Elemente-Methode” [FEM - Basic Principles and Applications of the Finite Element Method] Vieweg Verlag, 3rd Ed., 1999, from T. R. Chandrupalta and A. D. Belegundu: “Introduction to Finite Element in Engineering,” Prentice-Hall, 1991, and from DE 19927941 C1. Strength problems of all kinds, e.g., for stress distribution or stability, are solved numerically using a finite element simulation. For example, it is determined how a system made up of deformable objects is deformed and warped under external loads, and what stresses occur in the objects in these cases. The system may be the body of a motor vehicle, for example, and the objects may be components of the body. An electronic design model, available on computer, of a system to be studied is provided. The model preferably has the form of a three-dimensional CAD model.
In this model, a certain set of points, referred to as grid points, is established. Those surface or volume elements whose geometries are defined by the grid points are referred to as finite elements. The grid points form a mesh in the model; therefore, the procedure of establishing grid points and generating finite elements is referred to as meshing the model. The result of the procedure is known as a finite element mesh.
Curved surfaces or objects which are approximately considered as surfaces, e.g., the sheet metal of a motor vehicle body, are often split up into two-dimensional elements, also known as shell elements. The two-dimensional elements normally used are located in a single plane and have no curvatures. Roundings are approximated by a 90° edge, a 45° bevel or, less frequently, by a plurality of flat elements.
The most important types of two-dimensional elements are triangles having three grid points and quadrangles having four grid points. These three or four grid points also form the corners of the two-dimensional elements. In addition, quadrangles having eight or nine grid grid points are located at the midpoints of the quadrangle sides. One special type of finite element is the rigid object element, which connects grid points of different two-dimensional elements rigidly and with zero degrees of freedom. For example, a point at which a force acts upon a component (e.g., a threaded connection) is modeled by a rigid object element. No stress computation is performed for a rigid object element because this is impossible due to its stiffness. Different types of finite elements are available in “About MYSTRAN” at http://www.mystran.com/About_MYSTRAN.htm, consulted on Aug. 12, 2003.
Depending on the problem at hand, displacements of these grid points and/or rotations of the finite elements at these grid points and/or the stresses at certain points of these finite elements, namely in the points of integration, are introduced as unknowns. Equations are set up, which approximately describe displacements, rotations and/or stresses within a finite element.
Additional equations result from the relationships between different finite elements, e.g., from the fact that the principle of virtual work must be satisfied at the grid points and the calculated displacements must be continuous, and the boundary condition that no gaps or penetrations occur must be satisfied.
In many cases such equations are linear with respect to the unknowns. The finite element method may also be used, however, in the case of non-linear equations, for example, for equations in the form of polynomials. Non-linear equations occur, for example, in contact computations or heat radiation analyses. In general, a system of equations, which is often very extensive, is set up and solved numerically with the grid point displacements, grid point rotations, stresses in the points of integration or other quantities as unknowns. The solution of the equations describes, for example, the state of deformation of the system under given loads. Stress distributions, vibration responses, buckling responses, or service life predictions, for example, may be derived from this solution. For example, if the displacements of all grid points and stresses in the points of integration of a finite element are determined, the stress in the element may be derived from the material models.
For example, in the design of a motor vehicle body it should be predicted, using finite element simulation, what stresses occur in which areas of the body. Often different load scenarios are predefined and at least one simulation is carried out for each load scenario. During the simulation, the stresses in the points of integration of the finite elements are computed. From the stresses in its points of integration, the stress of a finite element is computed, for example, as the average of the stresses in the points of integration.
In a subsequent evaluation, those finite elements whose stresses exceed a predefined stress limiting value in the simulation are determined automatically. The positions of finite elements for which high stresses have been found in the finite element simulation indicate areas of the model in which the industrial system is at risk of undesirable deformations or damage, e.g., cracks, due to high stress.
The software tool MEDINA has functionalities for carrying out the meshing of a given model, i.e., for generating finite elements for the model. A description of MEDINA is found at http://www.c3pdm.com/des/products/medina/documentation/medina-D A4_e.pdf, consulted on Feb. 5, 2003. Using meshing, the type of finite element is determined for each finite element, i.e., for example, whether it is a triangle, a quadrangle, a rigid object element, or some other finite element. Furthermore, the spatial position of the finite element is determined. MEDINA provides a finite element mesh as the result of the meshing. A finite element tool inputs the generated finite element mesh and performs a finite element simulation, in which the stresses occurring in the finite elements are computed.
A finite element of the model is determined when it is a two-dimensional element and its stress exceeds a predefined stress limiting value. MEDINA then displays the result of the finite element simulation graphically; among other things, the determined finite elements are displayed in red in one representation of the model.
In H. Sandström and M. Shamlo: “Development of ADRIAN - Joint Analysis Software,” M.Sc. Thesis, Chalmers University of Technology, 2002, available at http://www.phi.chalmers.se/pub/msc/pdf/phimsc-sandstrom-shamlo-02.pdf, consulted on Jul. 21, 2003, a software program named ADRIAN is described. This program analyzes a finite element mesh of a motor vehicle body, in particular the palrtial models for w elded and other joints. ADRIAN analyzes, as MEDINA, the quality of the meshing and compares, for example, the aspect ratio, the taper, and the skew with threshold values.
A finite element having a stress that is greater than the stress limiting value may indicate an area of the industrial system described by the model which does not withstand the given load, and therefore may be a critical finite element. However, exceeding the predefined stress limiting value may also simply result from the fact that meshing of a model is inevitably an approximate model of the actual industrial system, for example, because the finite elements only approximately represent curvatures. Therefore, the finite element simulation only approximately reproduces reality, so that it is possible for an excess stress to occur just in the simulation, but not in reality.
U.S. Pat. No. 6,212,486 B1 describes a method for classifying finite elements having high stresses as critical. A finite element simulation is performed. A modal stress is determined for a finite element; the maximum possible ordinary stress and the shear stress are derived therefrom, and a maximum possible van Mises stress is computed. If this is greater than an upper limit, the finite element is critical. A fatigue analysis is performed using the variation of the stress of a finite element over time.
When using the methods known from the related art, it is left to the experience of an observer to evaluate the results of a finite element simulation, in particular the stresses in the finite elements. The related art provides no method that makes it possible to distinguish actual excess stresses from excess stresses resulting merely from the approximation of the model by finite elements.